This is a comprehensive introduction to the main concepts of abstract algebra, starting with group theory and continuing with ring theory, linear algebra, and fields. It would be appropriate for use in a single-semester introduction to abstract algebra course that covers the basic topics of group theory and ring theory. It would also be quite suitable for a two-semester algebra sequence.

One thing that stands out when reading this book is the quality of the explanations and expositions. Rotman does a very good job of motivating the topics and explaining things in common language. Students using this book would be well-served to have a strong background in discrete mathematics, as Rotman "hits the ground running" by establishing concepts including induction, greatest common divisors, and congruences in the first chapter. As with most texts on this subject, Rotman often pauses to ask open-ended questions when he is about to introduce a new topic. However, in this book these questions often lead to an interesting vignette or history lesson instead of just to the next theorem. With a subject as traditionally difficult as abstract algebra, it is nice for students to be able to see that the reason for proving a theorem is to solve a particular problem, rather than because it is simply the next theorem in the book.

As the book's title indicates, a large portion of the book is dedicated to applications. Unlike some books that tack applications onto the ends of sections, Rotman does a nice job of going into the details of the application, often developing specific notation and theorems. Some of my favorite applications are included, such as the 15-puzzle and using Burnside's lemma to solve counting problems. The attention to detail regarding the applications is apparent, and they do not seem "tacked on" as they often are in other texts.

One drawback of the book is that it is too comprehensive. There is so much material included that one could easily spend a week or two on each section. While Rotman does include some suggested syllabi, it would have been nice to have some more organizational insight given to the prospective instructor. If I were using this book, I would have to spend a large amount of time sorting through the sections ("Should I include Example 2.161?") in order to fit enough material into a 14 or 15-week semester. I would have preferred the book to be organized into a larger number of shorter sections, and a flowchart included so that I could better determine which sections are necessary and which applications and excursions can be omitted.

Overall, I liked Rotman's book, but I think one would have to be very dedicated to making it work within the confines of a semester or two. As many instructors know, well-constructed examples and applications often get discarded when we look at the calendar and see how much time is left in the semester compared with how much material is left to cover. Rotman's book contains great explanations, great examples, and great applications. Unfortunately, it just seems to have too much of everything, and not a lot of advice for the instructor on how to sort it all out.

James Hamblin is an Assistant Professor of Mathematics at Shippensburg University of Pennsylvania. His mathematical interests include origami, quilting, voting theory, and pretty much anything else he can get undergraduates interested in.